von Neumann’s mean ergodic theorem on complete random inner product modules

Abstract

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L_{ℱ}^p(ℰ,H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L^p(ℰ,H) (1≤p<∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), ℱ a sub σ-algebra of ℰ, and L_{ℱ}^p(ℰ,H) the complete random normed module generated by L^p(ℰ,H).

Key words： Random inner product module; random normed module; random unitary operator; random contraction operator; von Neumann’s mean ergodic theorem

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